There are many software packages you can use to do this for whichever platform you are using but the main two players are…
Itunes
The podcasts directory in the iTunes Store is where you find and subscribe to new podcasts:
The Podcasts section of your Library is where you manage your existing subscriptions and play the episodes you have downloaded:
To find and subscribe to a podcast:
After a few seconds, iTunes will begin to download the latest episode for the podcast you subscribed to. To listen to or watch the episode:
Podcast episodes in iTunes
You don’t need to have an account with an iTunes Store to get podcasts. However, you do need an account to submit a podcast to the iTunes Store.
iPodder (now called Juice)
This is a cross platform podcast download application
http://juicereceiver.sourceforge.net/
Why should I squeeze Juice?
The reason I use it is because it can be scheduled. Podcasts are relatively large files and so if your computer is going to trying to download podcasts whilst you are trying to work it will slow your Internet connection down. So I have set Juice to download al my latest podcasts in the middle of the night when I am asleep and so not using my computer! Juice can also import your podcasts into specific playlists within iTunes whilst I am asleep so when I sync up my iPod it brings in the latest podcasts as well.